| L(s) = 1 | − 2-s + 2.61·3-s + 4-s − 2.46·5-s − 2.61·6-s + 4.06·7-s − 8-s + 3.82·9-s + 2.46·10-s + 1.03·11-s + 2.61·12-s − 4.93·13-s − 4.06·14-s − 6.44·15-s + 16-s + 0.256·17-s − 3.82·18-s − 5.12·19-s − 2.46·20-s + 10.6·21-s − 1.03·22-s − 23-s − 2.61·24-s + 1.07·25-s + 4.93·26-s + 2.16·27-s + 4.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s − 1.10·5-s − 1.06·6-s + 1.53·7-s − 0.353·8-s + 1.27·9-s + 0.779·10-s + 0.310·11-s + 0.754·12-s − 1.36·13-s − 1.08·14-s − 1.66·15-s + 0.250·16-s + 0.0623·17-s − 0.902·18-s − 1.17·19-s − 0.551·20-s + 2.31·21-s − 0.219·22-s − 0.208·23-s − 0.533·24-s + 0.214·25-s + 0.967·26-s + 0.416·27-s + 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 - 0.256T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 0.0806T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 1.38T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 5.36T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79342719533994728533154271056, −7.48155619349967025507525392091, −6.82651490489515052630921420934, −5.46461641192417955352123862683, −4.50992566668611760628924732544, −4.01606665258633625839428955728, −3.07799574750335938295128165583, −2.16973232602760183981720825772, −1.61499123369325123840504706249, 0,
1.61499123369325123840504706249, 2.16973232602760183981720825772, 3.07799574750335938295128165583, 4.01606665258633625839428955728, 4.50992566668611760628924732544, 5.46461641192417955352123862683, 6.82651490489515052630921420934, 7.48155619349967025507525392091, 7.79342719533994728533154271056
